Something rather different in nature is going on when DxO performs barrel distortion corrections ???

Sure. DxO crops the image to a rectangular shape. If you ask kindly, I think LR will do the same. If additionally, you make sure that the amount of correction applied by the two is exactly the same, it might be an interesting comparison.

Well, the pixel-size is essentially the same (DxO has one more measly horizontal pixel). Are you suggesting that DxO Optics Pro up-samples their images by (heaven forbid) arbitrary amounts ?

Yes, there's simply no end to it.

Panini this, magister. Here in all it's glory is DxO's magical and mystically aligned Fisheye distortion correction at maximum setting. View the original at 20%, and you will see what William Blake saw.